International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 26-27
Section 1.1.4.10.5. Reduction of the number of independent components of fourth-rank polar tensors due to the symmetry of the strain and stress tensors^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
1.1.4.10.5. Reduction of the number of independent components of fourth-rank polar tensors due to the symmetry of the strain and stress tensors
Let us consider five examples of fourth-rank tensors:
In each of the equations from (1.1.4.9) to (1.1.4.10), the contracted product of a fourth-rank tensor by a symmetric second-rank tensor is equal to a symmetric second-rank tensor. As in the case of the third-rank tensors, this results in a reduction of the number of independent components, but because of the properties of the strain Voigt matrix, and because two of the tensors are endowed with intrinsic symmetry (the elastic tensors), the reduction is different for each of the five tensors. The above relations can be written in matrix form: where the second-rank tensors are represented by column matrices, which can each be subdivided into three submatrices and the matrix associated with the fourth-rank tensors is subdivided into nine submatrices, as shown in Section 1.1.4.9.1. The symmetry of the second-rank tensors means that submatrices 2 and 3 which are associated with them are equal.
Let us first consider the reduction of the tensor of elastic compliances. As in the case of the piezoelectric tensor, equation (1.1.4.9) can be written
The sums for have a definite physical meaning, but it is impossible to devise an experiment permitting and to be measured separately. It is therefore usual to set them equal in order to avoid an unnecessary constant:
Furthermore, the left-hand term of (1.1.4.11) remains unchanged if we interchange the indices i and j. The terms on the right-hand side therefore also remain unchanged, whatever the value of or . It follows that Similar relations hold for , , and : the submatrices 2 and 3, 4 and 7, 5, 6, 8 and 9, respectively, are equal.
Equation (1.4.1.11) can be rewritten, introducing the coefficients of the Voigt strain matrix: We shall now introduce a two-index notation for the elastic compliances, according to the following conventions: We have thus associated with the fourth-rank tensor a square matrix with 36 coefficients:
One can translate relation (1.1.4.12) using the matrix representing by adding term by term the coefficients of submatrices 2 and 3, 4 and 7 and 5, 6, 8 and 9, respectively:
Using the two-index notation, equation (1.1.4.9) becomes
A similar development can be applied to the other fourth-rank tensors , which will be replaced by matrices with 36 coefficients, according to the following rules.